Permutation Symmetry Research Articles

Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models.

We introduce the notion of mixed symmetry quantum phase transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector μ, when some Hamiltonian control parameters λ are varied. We use a three-level Lipkin-Meshkov-Glick model, with U(3) dynamical symmetry, to exemplify our construction. After reviewing the construction of U(3) unitary irreducible representations using Young tableaux and the Gelfand basis, we first study the case of a finite number N of three-level atoms, showing that some precursors (fidelity susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasiclassical) states of U(3) as variational states, we compute the lowest-energy density for each sector μ in the thermodynamic N→∞ limit. Extending the control parameter space by μ, the phase diagram exhibits four distinct quantum phases in the λ-μ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector μ=1 and shows a fourfold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schrödinger cat states.

Accurate Reduced-Cost CCSD(T) Energies: Parallel Implementation, Benchmarks, and Large-Scale Applications.

The accurate and systematically improvable frozen natural orbital (FNO) and natural auxiliary function (NAF) cost-reducing approaches are combined with our recent coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementations. Both of the closed- and open-shell FNO-CCSD(T) codes benefit from OpenMP parallelism, completely or partially integral-direct density-fitting algorithms, checkpointing, and hand-optimized, memory- and operation count effective implementations exploiting all permutational symmetries. The closed-shell CCSD(T) code requires negligible disk I/O and network bandwidth, is MPI/OpenMP parallel, and exhibits outstanding peak performance utilization of 50–70% up to hundreds of cores. Conservative FNO and NAF truncation thresholds benchmarked for challenging reaction, atomization, and ionization energies of both closed- and open-shell species are shown to maintain 1 kJ/mol accuracy against canonical CCSD(T) for systems of 31–43 atoms even with large basis sets. The cost reduction of up to an order of magnitude achieved extends the reach of FNO-CCSD(T) to systems of 50–75 atoms (up to 2124 atomic orbitals) with triple- and quadruple-ζ basis sets, which is unprecedented without local approximations. Consequently, a considerably larger portion of the chemical compound space can now be covered by the practically “gold standard” quality FNO-CCSD(T) method using affordable resources and about a week of wall time. Large-scale applications are presented for organocatalytic and transition-metal reactions as well as noncovalent interactions. Possible applications for benchmarking local CCSD(T) methods, as well as for the accuracy assessment or parametrization of less complete models, for example, density functional approximations or machine learning potentials, are also outlined.

Phase space theory for open quantum systems with local and collective dissipative processes

In this article we investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and local dissipators. Exploiting the permutation symmetry in our model, we employ a phase space approach for the solution of this equation in terms of a diagonal representation with respect to certain generalized spin coherent states. Remarkably, this allows to interpolate between mean field theory and finite system size in a formalism independent of Hilbert-space dimension. Moreover, in certain parameter regimes, the evolution equation for the corresponding quasiprobability distribution resembles a Fokker–Planck equation, which can be efficiently solved by stochastic calculus. Then, the dynamics can be seen as classical in the sense that no entanglement between the two-level systems is generated. Our results expose, utilize and promote techniques pioneered in the context of laser theory, which can be powerful tools to investigate problems of current theoretical and experimental interest.

Modern State of the Pauli Exclusion Principle and the Problems of Its Theoretical Foundation

The Pauli exclusion principle (PEP) can be considered from two aspects. First, it asserts that particles that have half-integer spin (fermions) are described by antisymmetric wave functions, and particles that have integer spin (bosons) are described by symmetric wave functions. It is called spin-statistics connection (SSC). The physical reasons why SSC exists are still unknown. On the other hand, PEP is not reduced to SSC and can be consider from another aspect, according to it, the permutation symmetry of the total wave function can be only of two types: symmetric or antisymmetric. They both belong to one-dimensional representations of the permutation group, while other types of permutation symmetry are forbidden. However, the solution of the Schrödinger equation may have any permutation symmetry. We analyze this second aspect of PEP and demonstrate that proofs of PEP in some wide-spread textbooks on quantum mechanics, basing on the indistinguishability principle, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of wave function. So, it cannot be used as a criterion for the PEP verification. However, as follows from our analysis of possible scenarios, the permission of states with permutation symmetry more general than symmetric and antisymmetric leads to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature particles only in symmetric and antisymmetric permutation states is not accidental, since all symmetry options for the total wave function, except the antisymmetric and symmetric, cannot be realized. From this an important conclusion follows, we may not expect that in future some unknown elementary particles that are not fermions or bosons can be discovered.

Multiple positive solutions for coupled Schrödinger equations with perturbations

For coupled Schrodinger equations with nonhomogeneous perturbations we give several results on the existence of multiple positive solutions. In particular in one case we consider perturbations of the permutation symmetry.

Phases and quantum phase transitions in an anisotropic ferromagnetic Kitaev-Heisenberg- Γ magnet

We study the spin-$1/2$ ferromagnetic Heisenberg-Kitaev-$\Gamma$ model in the anisotropic (Toric code) limit to reveal the nature of the quantum phase transition between the gapped $Z_2$ quantum spin liquid and a spin ordered phase (driven by Heisenberg interactions) as well as a trivial paramagnet (driven by pseudo-dipolar interactions, $\Gamma$). The transitions are obtained by a simultaneous condensation of the Ising electric and magnetic charges-- the fractionalized excitations of the $Z_2$ quantum spin liquid. Both these transitions can be continuous and are examples of deconfined quantum critical points. Crucial to our calculations are the symmetry implementations on the soft electric and magnetic modes that become critical. In particular, we find strong constraints on the structure of the critical theory arising from time reversal and lattice translation symmetries with the latter acting as an anyon permutation symmetry that endows the critical theory with a manifestly self-dual structure. We find that the transition between the quantum spin liquid and the spin-ordered phase belongs to a self-dual modified Abelian Higgs field theory while that between the spin liquid and the trivial paramagnet belongs to a self-dual $Z_2$ gauge theory. We also study the effect of an external Zeeman field to show an interesting similarity between the polarised paramagnet obtained due to the Zeeman field and the trivial paramagnet driven the pseudo-dipolar interactions. Interestingly, both the spin liquid and the spin ordered phases have easily identifiable counterparts in the isotropic limit and the present calculations may shed insights into the corresponding transitions in the material relevant isotropic limit.

The Pauli Exclusion Principle and the Problems of its Theoretical Substantiation1

The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrodinger equation may have any permutation symmetry. It is demonstrated that the proof in some textbooks on quantum mechanics that only symmetric and antisymmetric states can exist is wrong. However, the scenarios, in which arbitrary permutation symmetry (degenerate permutation states) is permitted, lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature of particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate should not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in the future some unknown elementary particles can be discovered that are not fermions or bosons.

Crossover exponents, fractal dimensions and logarithms in Landau\u2013Potts field theories

We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry S_q in d=6-epsilon (Landau–Potts field theories) and d=4-epsilon (hypertetrahedral models) up to three loops. We use our results to determine the epsilon -expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests (qrightarrow 0), and bond percolations (qrightarrow 1). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of q upon analytic continuation, which are linked to logarithmic corrections of CFT correlators, and use the epsilon -expansion to determine the universal coefficients of such logarithms.

Permutation Symmetry in Coherent Electrons Scattering by Disordered Media

A non-Anderson weak localization of an electron beam scattered from disordered matter is considered with respect to the principle of electron indistinguishability. A weak localization of electrons of a new type is essentially associated with inelastic processing. The origin of inelasticity is not essential. We take into account the identity principle for electron beam and electrons of the atom of the scatterer with an open shell. In spite of isotropic scattering by each individual scatterer, the electron exchange contribution has a hidden parameters effect on the resulting angular dependence of the scattering cross-section. In this case, the electrons of the open shell of an atomic scatterer can be in the s-state, that is, the atomic shell remains spherically symmetric. The methods of an invariant time-dependent exchange perturbation theory and a Green functions with exchange were applied. An additional angular dependence of the scattering cross-section appears during the coherent scattering process. It is shown exactly for the helium scatterer that the role of exchange effects in the case of a singlet is negligible, while for the triplet state, it is decisive, especially for those values of the energy of incident electrons when de Broglie’s waves are commensurate with the atomic.

Quantizing the propagated field through a dielectric including general class of permutation symmetries for nonlinear susceptibility tensors

In this paper, we present a quantum theory for field propagation through a three-dimensional dielectric, when the frequency permutations of nonlinear susceptibility tensors should not be ignored. This treatment is important for the processes in which multiple frequencies are generated. The Hamiltonian of our system is obtained by introducing a unique Lagrangian and making use of the dual potential. This system is quantized by imposing the Dirac commutation relations. Finally, appropriate creation and annihilation operators are defined to derive the nonlinear Hamiltonian operator under frequency permutations of the nonlinear susceptibility tensors. As an example, our simulation results show the impact of frequency permutations on pulse propagation along a one-dimensional simple waveguide.

High-Performance, Graphics Processing Unit-Accelerated Fock Build Algorithm.

We present a high-performance, GPU (graphics processing unit)-accelerated algorithm for building the Fock matrix. The algorithm is designed for efficient calculations on large molecular systems and uses a novel dynamic load balancing scheme that maximizes the GPU throughput and avoids thread divergence that could occur due to integral screening. Additionally, the code adopts a novel ERI digestion algorithm that exploits all forms of permutational symmetry, combines efficiently the evaluation of both Coulomb and exchange terms together, and eliminates explicit thread synchronization requirements. Performance results obtained using a number of large molecules reveal remarkable speedups up to 24.4× with respect to the QUICK GPU code and up to 237× with respect to the GAMESS CPU parallel code.

Eight-Dimensional Wave Packet Dynamics Within the Quantum Transition-State Framework: State-to-State Reactive Scattering for H2 + CH3 ⇆ H + CH4.

The present work investigates the calculation of S-matrix elements for six-atom reactions combining reduced-dimensional wave packet dynamics and the quantum transition-state framework. We employ the eight-dimensional (8D) model Hamiltonian developed by Palma and Clary [J. Chem. Phys. 2000, 112, 1859-1867] and reduce basis set sizes as well as the number of wave packets by exploiting space inversion and permutation symmetry. Mode-specific chemistry in the H2 + CH3 ⇆ H + CH4 reaction is studied with full quantum-state resolution. Results for the H + CH4 reaction are compared to full-dimensional benchmark results. Detailed state-to-state results for the H2 + CH3 reaction are presented for the first time. Although the "loss of memory" effect dominates for total energies up to 0.6 eV, more complex patterns emerge at higher energies. The agreement between the present reduced-dimensional and the accurate full-dimensional results is generally good. However, shortcomings in the reduced-dimensional model can also be noted. They are related to the description of the symmetric and asymmetric C-H stretch motion in the CH4 molecule.

Targeted free energy estimation via learned mappings.

Free energy perturbation (FEP) was proposed by Zwanzig [J. Chem. Phys. 22, 1420 (1954)] more than six decades ago as a method to estimate free energy differences and has since inspired a huge body of related methods that use it as an integral building block. Being an importance sampling based estimator, however, FEP suffers from a severe limitation: the requirement of sufficient overlap between distributions. One strategy to mitigate this problem, called Targeted FEP, uses a high-dimensional mapping in configuration space to increase the overlap of the underlying distributions. Despite its potential, this method has attracted only limited attention due to the formidable challenge of formulating a tractable mapping. Here, we cast Targeted FEP as a machine learning problem in which the mapping is parameterized as a neural network that is optimized so as to increase the overlap. We develop a new model architecture that respects permutational and periodic symmetries often encountered in atomistic simulations and test our method on a fully periodic solvation system. We demonstrate that our method leads to a substantial variance reduction in free energy estimates when compared against baselines, without requiring any additional data.

Second-order nonlinear optical properties of [formula omitted]-shaped pyrazine derivatives

The linear and nonlinear optical (NLO) properties of a series of Λ-shaped derivatives containing a 4,5-dicyanopyrazine acceptor unit, N,N-dimethylamino donor groups and systematically enlarged π-conjugated linkers are investigated by means of UV/Visible and Hyper-Rayleigh scattering spectroscopies. Density functional theory calculations are also carried out to rationalize the magnitude and symmetry of the NLO responses. The results demonstrate that these compounds possess two low-lying excited electronic states close to each other in energy, which are accessible through one-photon optical transitions respectively polarized perpendicular and parallel to the two-fold molecular axis. These two states contribute additively to the first hyperpolarizability, which exhibits a dominant dipolar character. We also show that the NLO responses significantly deviate from Kleinman's index permutation symmetry.

Irreducible Cartesian Tensor Analysis of Harmonic Scattering from Chiral Fluids

Symmetry principles of several distinct kinds are revealingly engaged in an analysis focussing on third harmonic scattering, a current focus of research on nonlinear optics in chiral media. Analysis in terms of irreducible Cartesian tensors elucidates the detailed electrodynamical origin and character of the corresponding material properties. Considerations of fundamental charge, parity and time reversal (CPT) symmetry reveal the conditions for an interplay of transition multipoles to elicit a chiral response using circularly polarised pump radiation, and the symmetry of quantised angular momentum underpins the associated selection rules and angular distribution. The intrinsic structural symmetry of chiral scatterers determines their capacity to exhibit differential response. Exploiting permutational index symmetry in the response tensors enables quantitative assessment of the boundary values for experimentally measurable properties, including circular intensity differentials.

Perfect quantum state transfer on diamond fractal graphs

In the quest for designing novel protocols for quantum information and quantum computation, an important goal is to achieve perfect quantum state transfer for systems beyond the well-known one- dimensional cases, such as 1D spin chains. We use methods from fractal analysis and probability to find a new class of quantum spin chains on fractal-like graphs (known as diamond fractals) which support perfect quantum state transfer and which have a wide range of different Hausdorff and spectral dimensions. The resulting systems are spin networks combining Dyson hierarchical model structure with transverse permutation symmetries of varying order.

Mirror symmetry of nonabelian Landau–Ginzburg orbifolds with loop type potentials

We study the bigraded vector space structure of Landau–Ginzburg orbifolds with the permutation symmetries of variables. We calculate the formula of the Poincaré polynomial for a certain loop polynomial orbifoldized by a semidirect product of a diagonal symmetry group and cyclic permutation group and check that the generalization of Berglund–Hübsch transpose gives mirror symmetric pairs for these cases.

A general code for fitting global potential energy surfaces via CHIPR method: Direct-Fit Diatomic and tetratomic molecules

A general program to fit global adiabatic potential energy surfaces of up to tetratomic molecules to ab initio points and available spectroscopic data for simple diatomics is reported. It is based on the Combined-Hyperbolic-Inverse-Power-Representation (CHIPR) method. The final form describes all dissociating fragments and long-range/valence interactions, while obeying the system permutational symmetry. The code yields as output a Fortran 90 subroutine that readily evaluates the potential and gradient at any arbitrary geometry. Program summaryProgram title: CHIPR-4.0CPC Library link to program files:http://dx.doi.org/10.17632/8wdv87gt5x.2Licensing provisions: GPLv3Programming language: Fortran 90Journal reference of previous version: C. M. R. Rocha, A. J. C. Varandas, Comput. Phys. Commun. 247 (2020) 106913Does the new version supersede the previous version?: YesReasons for the new version: CHIPR-4.0 is a major extension of the previous CHIPR-3.0 code [1] which, besides diatomics and triatomics, also includes the possibility of fitting global adiabatic potential energy surfaces (PESs) of A4-, AB3-, A2B2-, ABC2- and ABCD-type tetratomic molecules [3]. Additionally, a new feature is that it allows the user to fine-tune ab initio diatomic curves using available spectroscopic data.Summary of revisions:• Implementation of CHIPR’s polynomial form and permutation operators for tetratomic molecules• Implementation of a direct-fit module to fine-tune ab initio potential energy curves using spectroscopic data• Automatic global minimum search on the diatomic, triatomic or tetratomic final PESs• Automatic harmonic vibrational analysis at the optimum molecular geometriesNature of problem: This version of the CHIPR code, CHIPR-4.0, provides a set of subroutines to fit global adiabatic potential energy surfaces of up to tetratomic molecules using ab initio and (for diatomics) optional fine-tuning experimental data.Solution method: CHIPR-4.0 uses the Combined-Hyperbolic-Inverse-Power-Representation (CHIPR) [2] method to interpolate and extrapolate ab initio data points.Additional comments including restrictions and unusual features: For triatomic and tetratomic fits, the user must supply subroutines containing the analytic forms defining the sum of two- and two-plus-three-bodies, respectively. These are utilized internally to calculate (from ab initio data) the corresponding three- and four-body energies to be fitted. The current version of this code is strictly based on single-sheeted analytical PESs. High-accuracy in diatomic refinements is currently limited to the 1Σ single-curve case.

The Method of High Accuracy Calculation of Robot Trajectory for the Complex Curves

Abstract The geometric model accuracy is crucial for product design. More complex surfaces are represented by the approximation methods. On the contrary, the approximation methods reduce the design quality. A new alternative calculation method is proposed. The new method can calculate both conical sections and more complex curves. The researcher is able to get an analytical solution and not a sequence of points with the destruction of the object semantics. The new method is based on permutation and other symmetries and should have an origin in the internal properties of the space. The classical method consists of finding transformation parameters for symmetrical conic profiles, however a new procedure for parameters of linear transformations determination was acquired by another method. The main steps of the new method are theoretically presented in the paper. Since a double result is obtained in most stages, the new calculation method is easy to verify. Geometric modeling in the AutoCAD environment is shown briefly. The new calculation method can be used for most complex curves and linear transformations. Theoretical and practical researches are required additionally.

Permutationally Invariant, Reproducing Kernel-Based Potential Energy Surfaces for Polyatomic Molecules: From Formaldehyde to Acetone

Constructing accurate, high-dimensional molecular potential energy surfaces (PESs) for polyatomic molecules is challenging. Reproducing kernel Hilbert space (RKHS) interpolation is an efficient way to construct such PESs. However, RKHS interpolation is computationally most effective when the input energies are available on a regular grid. Thus, the number of reference energies required can become very large even for pentaatomic systems making such an approach computationally prohibitive when using high-level electronic structure calculations. Here, an efficient and robust scheme is presented to overcome these limitations and is applied to constructing high-dimensional PESs for systems with up to 10 atoms. Using energies as well as gradients reduces the number of input data required and thus keeps the number of coefficients at a manageable size. The correct implementation of permutational symmetry in the kernel products is tested and explicitly demonstrated for the highly symmetric CH4 molecule.